# Properties, Laws and Formulae

## Real Numbers

### Properties of Real Numbers

For any real numbers a, b and c;

$$\forall \; a, b \in R \quad \Rightarrow \quad a+b \in R$$

• Closure Property w.r.t. Multiplication

$$\forall \; a, b \in R \quad \Rightarrow \quad a . b \in R$$

$$(a + b) + c = a + (b + c)$$

• Associative Property w.r.t. Multiplication

$$(a . b) . c = a . (b . c)$$

There is a unique number zero (0) called additive identity such that:

$$a + 0 = 0 + a = a$$

• Multiplicative Identity Property

There is a unique number one (1) called multiplicative identity such that:

$$a . 1 = 1 . a = a$$

For any real number a, there exists a unique real number −a called additive inverse of a such that:

$$a + (−a) = 0 = (−a) + a$$

• Multiplicative Inverse Property

If a $a \neq 0$ is a real number, there exists a unique real number $1 \over a$ called multiplicative inverse of a such that:

$$a . {1 \over a} = 1 = {1 \over a} . a$$

$0 \in R \quad$ has no multiplicative inverse

$$a + b = b + a$$

• Commutative Property w.r.t. Multiplication

$$a . b = b . a$$

• Distributive Property of Multiplication over addition

$$a (b + c) = a . b + a . c$$

or

$$(a + b) c = a . c + b . c$$

## Percentage and Financial Arithmetic

$$Profit \;\; percentage = {profit \over cost \;\; price} \times 100$$

$$Loss \;\; percentage = {loss \over cost \;\; price} \times 100$$

$$Discount = Marked \;\; price \; − \; Selling \;\; price$$

$$Percentage \;\; discount = {discount \over marked \;\; price} \times 100$$

$$Profit \;\; / \;\; Markup = Principal \times Rate \times Time$$

$$i.e. \;\;\; I \; = \; P \; R \; T$$

$$Profit \; / \; Markup \; Rate = \; {Markup \times 100 \over Principal \times Time}$$

## Number Sequence and Patterns

For finding nth term or general term of an arithmetic sequence,

$$a_n = a_1 + (n − 1) d$$

For finding next terms by term to term rule in arithmetic sequence,

$$a_n = a_{n−1} + d$$

For finding nth term or general term of a geometric sequence,

$$a_n = a_1 \; r^{n − 1}$$

## Expansion and Factorization

### Basic Algebraic Formulas

$$(a + b)^2 = a^2 + 2ab + b^2$$

$$(a − b)^2 = a^2 − 2ab + b^2$$

$$a^2 − b^2 = (a + b)(a − b)$$

#### Manipulation of Algebraic Expression

$$(a + b)^3 = a^3 + 3ab (a + b) + b^3$$

$$(a − b)^3 = a^3 − 3ab (a − b) − b^3$$

### Laws of Exponents

In general, for any non-zero integers x and y where m and n are whole numbers.

#### Law of Sum of Powers

$$x^m \times x^n = x^{m + n}$$

#### Law of Quotient of Powers with same base

$$x^m \div x^n = x^{m − n}$$

#### Law of Power of Power

$$(x^m)^n = x^{m n}$$

#### Law of Power of the Product

$$x^m \times y^m = (x \times y)^m$$

#### Law of Power of the Quotient

$$x^m \div y^m = \left( {x \over y} \right)^m$$

#### Law of Zero Power

$$x^0 = 1$$

#### Law of Negative Power

$$x^{−m} = {1 \over x^m}$$

#### Laws of Fractional Exponents

$$x^{1 \over n} = \sqrt[n] x$$

$$x^{m \over n} = ( x^m )^{1 \over n} = \sqrt[n] {x^m}$$

$$x^{m \over n} = \left( x^{1 \over n} \right) ^m = \left( \sqrt[n] x \right)^m$$